Alfvén Instability in Magnetic Flux Tubes with Isothermal Flow

1. Alfvén Instability in Magnetic Flux Tubes with Isothermal Flow IAFA 2011, Alpbach, Austria Youra Taroyan yot@aber.ac.uk Institute of Mathematics and Physics, AberystwythUniversity, UK

2. Observational context • SUMER study of chromospheric and transition region line profiles (Peter, 2000, 2001) • Double Gaussian fit needed for line profiles in bright network structures: core and tail components

3. Observational context Peter (2001) • Core component corresponds to small closed loops • Blue-shifted tail component corresponds to funnels that are connected to the corona • Non-thermal broadenings of the tail component  non-linear Alfvén waves

4. Observational context Hinode/EISmeasurements ofsubsonic upflows of tens of km/s and enhanced nonthermal velocities near the footpoints of active region loops (Hara et al. 2008)

5. Observational context Tian, Marsch et al. (2009)

6. Observational context Jess et al. (2009) studied H-alpha absorption profiles with SST and found FWHM oscillations with an amplitude of 3 km/s accompanied by a blueshift of 23 km/s.

7. Observational context • SDO/Hinode observations by De Pontieu et al. (2009, 2011) show ubiquitous mass supply from chromosphere to corona • Plasma in fountainlike jets or spicules heated to transition region / coronal temperatures • Upflow speeds ~ 100km/s, heights ~ 10-20Mm • Earlier studies by De Pontieu et al. (2007) associate similar events with transverse waves

8. Examples of other similar observations:Xia et al. (2003, 2004), McIntosh (2009, 2011) … • What is the relationship between the magnetic flux tube geometry and the flow? • Why are the observed upflows associated with nonthermal line widths? • How significant is the contribution of broadening to chromospheric/coronal heating? Questions

9. B0 u0 r s=0 g

10. B0 x=0 x=L A ‘simple’ model

11. Stability analysis • Apply t -> ω Laplace transform • Connect the solutions in the + and – regions at x=L • Invert and determine the response of the system to an arbitrary perturbation • Response depends on the location of singularities in the complex ω plane • Location of singularities depends on the sign of

12. Case 1: incompressible flow

13. Case 2: compressible flow

14. Case 3: compressible flow

15. Conclusions from the analytical model • An instability exists when the flow is compressible enough • No shear required • Sub-sonic and sub-Alfvenic flow • Taroyan, PRL 2008

16. - + Corona s=0 s=L Taroyan, ApJ 2009

17. B0 u0 Horizontal flux tubes with isothermal flow r s=0

18. Stability analysis • Divide the tube into two parts: variable flow for 0<s<L and constant flow for s>L • Fourier transform the equations for axisymmetric twists • Find numerical solutions in 0<s<L and analytical solutions in s>L • Connect the solutions at x=L • Solve the resulting numerical dispersion relation and find the complex frequencies, i.e., determine stability of the system to an arbitrary twist

19. Conclusion: horizontal flux tubes are unstable when expansion (flow deceleration) is rapid enough!

20. B0 u0 Vertical flux tubes with isothermal flow r s=0 g

21. Stability analysis • Divide the tube into two parts: variable flow for 0<s<L and constant flow for s>L • Fourier transform the equations for axisymmetric twists • Solutions in s>L expressed in terms of hypergeometric functions. Select the one that remains finite at the Alfven point (flow speed = Alfven speed). • Apply the shooting method to determine stability of the system to an arbitrary twist

24. stable unstable stable

25. Conclusions • Isothermal flux tubes with smooth flow profiles can be unstable with respect to linear torsional perturbations • The instability arises in both horizontal and vertical flux tubes. • Amplification factor of 100 in about 10 min • The presented instability mechanism offers a straightforward explanation for the observed nonthermal broadenings associated with upflows in magnetic regions of the solar atmosphere